I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis. He believed in the method of letting students see how a mathematician's mind works, so he came to the lectures largely unprepared, and was proving theorems as they were met in the textbook. Consequently he would not always follow the straightest path to the destination. Occasionally he ran afoul full speed, and had to backtrack a bit or even start over.

Does this method help a large enough group of students to make it worth our while?

My views in what follows.

Pros

Students do get to see how a trained mathematician approaches the task of proving a theorem.

They learn that doing math is not a single-track exercise.

Students see the real way of doing math as opposed to fully digested polished presentation in the book.

This slows down the progress a bit, which may help some.

Cons

Taking notes (if a student chooses to do so meticulously) becomes a nightmare, when the teacher is hopping from one step to another

This slows down the progress a bit making it harder to cover all the material.

Doesn't invite student participation (unless you are willing to invest even more time on it).

What I have tried/done

Accidentally doing this (krhm).

After a motivational example I go straight to the proof of the main result of the day's lecture. I keep adding the necessary assumptions to the statement of the theorem as we encounter them. Also I get these lightbulb moments "Ahh, this is why the colleague who wrote these lecture notes had that lemma a few pages back. Comes in handy here." Naturally I had skipped that lemma, but go through it now - distracting some, but also giving a more realistic impression of how math gets done.

Summary of the discussion

The views expressed by participants in this discussion converge to AFAICT:

Letting students see dead-ends of trains of thought has, indeed, a lot of pedagogical merit, and could be turned into a tool of inviting student participation.

But these wrong turns should be crafted as carefully as the polished examples. They should not be random by-products of un(der)preparation.

Good teachers get their results in the old-fashioned way of earning them.

All: Thanks for sharing your ideas and suggestions in how to best implement this in practice.

13 Answers
13

Yes, showing your students that you too can make mistakes, and that real mathematics is not a linear process, are both very good ideas.

However, this does not mean that you should come to the lecture unprepared!

If I seem a bit vehement about this, it's because I've met all too many math lecturers who seem to feel a need to "prove their manlyhood" (or at least their awesome math skills) by refusing to prepare their proofs ahead of the lecture and instead proving everything on the spot. Trust me, spending 30 minutes out of a two-hour lecture watching the lecturer look for a sign error they made 15 minutes earlier gets old real fast.

The problem is that any lecture time you spend backtracking and correcting errors is time you could've spent teaching the students something new. I've never heard of a lecturer complaining about having too much time to cover everything they'd like, so good lecturers usually need to carefully budget their time, and any time spent finding and fixing mistakes in front of the class takes time out of this budget that has to be made up somehow.

Also, while showing your students a carefully planned example of how a mathematician would really approach a new problem can be very educational, that's not the same as just having the students twiddle their thumbs while you stare at the blackboard in confusion, which is what's more likely to happen if you actually get stuck in the middle of a proof.

Instead, here's a few ideas you could try, if you think your lectures are too polished and would benefit from more mistakes and spontaneity:

Fake it: Prepare your lectures as carefully as you can, but make occasional deliberate mistakes when writing on the blackboard. (You could even write something like "make mistake here" in your notes.) Encourage your students to try and spot these mistakes (with small rewards, if necessary). If they don't spot them in a few minutes, pretend to notice them yourself and fix them.

Of course, this also helps encourage your students to spot any genuine mistakes you might make.

Make it an event: Pick a specific problem that you're fairly sure you can solve on the spot, and allocate some extra time for it. Make sure to tell your students that you haven't prepared notes for that specific proof in advance — this also has the useful effect of helping your students realize that you do normally prepare your proofs in advance.

Make sure to keep describing what you're thinking while solving the problem out loud, and encourage your students to offer suggestions if you get stuck. Optionally, prepare a "cheat sheet" in case you really do get stuck, or pick the problem so that you can leave completing the proof as a bonus exercise.

Act it out: Same as above, but actually have a plan for your "spontaneous" proof. That is, when preparing the lecture, prove the theorem in advance, but write down not just the final "tidy" solution, but also a log of any mistakes and dead ends you encounter. Then re-enact the full proof, dead ends and all, in front of the class.

Even if you don't feel like actually re-enacting your solving process in every detail, you can still include some deliberate dead ends in your lecture plan. For example, once you've shown how to prove the stability of a stable node using phase-plane analysis, try doing it for a stable focus instead. Or pick a pumpable non-regular language and try to prove its nonregularity using the pumping lemma, before giving up and using Myhill–Nerode instead.

Also, even if you don't feel like using any such tricks, one thing you can do to make your usual "tidy" proofs seem a bit less linear is to sprinkle them with details about possible dead ends ("The obvious way to try and prove this would be to use lemma 1, but that doesn't work here, because...") or historical notes ("The original proof of this theorem was 50 pages long; the short proof I'll show you was only found 38 years later by...").

$\begingroup$Quite a nice first post indeed -- hope to see more of you around here!$\endgroup$
– Chris Cunningham♦Apr 13 '14 at 17:42

$\begingroup$Thanks for sharing your thoughts, Ilmari. Particularly useful to me as a fellow Finn, because you have been exposed to the culture of our university math education. I have been uncertain about "faking it", because I don't trust my acting ability. May be it being fake is not a problem unless I overdo it. Mind you, it would not occur to me to present a numerical example unprepared. Intermediate value theorem OTOH...$\endgroup$
– Jyrki LahtonenApr 14 '14 at 11:32

3

$\begingroup$Absolutely! Completely agree with this. I'd add that "faking" doesn't have to mean "acting". You can be completely honest about the fact that you're faking. The key is that you, the lecturer, should be in control at all times (as far as possible) and that making actual mistakes means losing control, but there are ways of achieving the same ends without losing control and this answer gives suggestions on that.$\endgroup$
– Andrew StaceyApr 15 '14 at 19:23

Here is something that you will hear from students who attend clear, perfectly-prepared lectures:

"When I'm in class watching you do it, it all makes perfect sense and seems really easy. But then I go home and work on it and I don't know how to do anything."

This is a disastrous outcome in introductory courses because it makes your students think that you are smart and they are dumb.

So in my opinion, even in a non-proofs class, a lecturer must do all of these:

Make mistakes, cross them out in red, and fix them

Stop, acting puzzled, at a place where a student might get stuck

Show what it looks like when a technique is being misapplied

Basically the goal is to give the students' brains something to process during the lecture.

I'd also say that some of this actually does invite more class participation:

If every single time you solve $x^2 = C$ you make the "mistake" of only finding one solution, it becomes a kind of game to the students to notice the error first.

When the correct answer to "what should I do next" is sometimes "give up on this approach??," it takes a lot of pressure off of students who are used to the airwaves only being filled with perfectly correct reasoning.

$\begingroup$Thanks for sharing your thoughts. Close to what my thinking was before reading the other answers. The others haven't really changed it that much, but rather they supplemented it with some valid concerns. I'm afraid I can't use that $x^2=C$ example as I take it as a dogma from week one not to forget the difference between one-way implication and equivalence. But I can find other similar catches :-)$\endgroup$
– Jyrki LahtonenApr 14 '14 at 13:24

What a good question this is. Others have already given good answers. I think Ilmari Karonen's answer is closest to my own heart. For now let me just respond to one aspect of the question:

I have heard the story (may be an urban legend?) of a top professor who occasionally wanted to teach freshman analysis. He believed in the method of letting students see how a mathematician's mind works, so he came to the lectures largely unprepared, and was proving theorems as they were met in the textbook. Consequently he would not always follow the straightest path to the destination. Occasionally he ran afoul full speed, and had to backtrack a bit or even start over.

I find this "top professor"'s teaching practice obnoxious. That's a matter of opinion, but the proposed justification contains (perhaps intentionally!!) several points which I find fallacious. Namely:

$\bullet$ I believe in letting my students see how a mathematician's mind works too. But whenever a mathematician is involved in teaching a course, one sees, in one way or another, how a mathematician's mind works. The idea that under-preparation gives a truer view to the mathematician's mind is ridiculous. Much of what a mathematician does is to think things out in advance and take advantage of what she has figured out.

$\bullet$ "...and were proving the theorems largely as they were met in the textbook" Already a problem: in closely following a textbook students are -- at best -- seeing how the textbook author's mind works, not the instructor's. In a course which consists primarily of theorems and proofs (I wonder in what land "freshman analysis" is such a course; it is not the United States, certainly; we rarely teach analysis in any form to freshmen), strictly adhering to the logical sequence of the textbook turns the lecture into a dramatic reading of the textbook. This is so much less than what an instructor should be doing in class.

$\bullet$ Running afoul at full speed and having to backtrack or start over entirely are reasonable classroom events. To see them happen at least once in a given course is probably a good thing: I have heard students react surprisingly (to me) positively when this sort of thing happens in my own courses. But as others have pointed out, when it happens once or twice it is heartening. When it causes an entire lecture to become derailed and then repeated the next time without a fatal sign mistake, it is clearly not good use of class time, and it will be frustrating to students.

The other fallacy here is that one needs to take such a completely derelict "let it all hang out" lecturing attitude in order for such mistakes to occur in the lecture. I have been teaching for many years, and I still make mistakes in my calculus lectures, let alone in graduate courses in which I am presenting the material for the first time. A certain positive rate of errors seems all but inevitable: if I asked you to prepare complete lecture notes, got a team of experts to check them over, gave them back to you and repeated until we were satisfied that they are seamlessly correct and complete, and then allowed you to bring the notes into the lecture and give a lecture which consisted purely of writing the notes on the board, then you will still make errors a positive percentage of the time. (In fact, as one learns, there is an interior maximum for this with respect to preparation: if you prepare too much then you stop thinking about what you're doing, and that makes errors more likely to occur.) The idea that one needs to make sure to be unprepared so as to get a desirable density of errors is really risible to me.

Let me close by saying that the pedagogical practice of intentional mistakes is a good one, and I have used it myself: if I want to introduce a subtle point in an advanced course, I will often introduce it by doing an example or a computation in which this subtle point is ignored and a contradiction is reached. I think this is an excellent example of a "mathematician's mind at work". The difference is that I have nevertheless prepared this in advance and the mistake is intentional. (At least, the intended mistake is intentional! I remember that earlier this semester I gave an argument in which I warned in advance that I was going to make a mistake in the proof. After the lecture was over, someone came to me and pointed out that I seemed to assume what I wanted to prove. I smacked myself in the head and wondered how I could have done something so terrible. She responded: "Well, you said there would be a mistake.")

$\begingroup$Thanks for sharing your thoughts, Pete. My choice of phrase in the background story may have been somewhat unfortunate in that it made the professor appear to be very disrespectful towards his students. Well, unfortunately I have this tendency to be overly dramatic. Your last paragraph I agree with 100%, and will try that something like that some time.$\endgroup$
– Jyrki LahtonenApr 14 '14 at 11:28

Getting students engaged in the act of catching and repairing mistakes in proofs and other solutions to problems can have a great deal of educational value. Lara Alcock has written a lot about "proof validation" and its role in undergraduate math education, for example in this paper, and there's quite a few others who also have done research on this. The tl;dr version is that activities that ask students to decide whether a proof is valid, and then find and repair the error if there is one, significantly improve students' abilities to then go and construct correct proofs on their own.

However: There is a huge difference between setting up intentional, well-designed proof validation activities one the one hand, and doing what this apocryphal professor did on the other hand, which is to come into class and start a proof cold without any idea where s/he is going.

For one thing, in the urban legend case the professor is still the center of attention, not the student; the professor surely can contribute to helping students understand the perspective of an expert problem-solver. But it's probably best to wait until after the students have had a chance to do it on their own, so that students will have a basis for paying attention to how the professor does it.

For another thing, it just seems like carelessness to go into a class unprepared. Basically you should never do that, no matter how good of a mathematician you are, out of professionalism.

Finally, it can backfire. I've seen situations where professors try this sort of thing and then get hopelessly lost, and they lose the students' trust. And once you've lost that, it is very hard to gain it back, and learning is hard when you don't trust the prof.

So -- rather than take a freewheeling approach, design good proof activities that target specific misconceptions or decision-making processes you'd like to bring out in class, then hand those to students and let them evaluate the proofs firsthand.

Here's an anecdote from being an undergraduate student working in a tutorial center. Students could come in and ask us questions about a fairly long list of standard introductory courses (we each wore badges indicating which courses we were (hopefully) competent to help with). I was terrified to do this at first, because I figured I'd be asked to help with lots of problems I wouldn't remember how to do (having actually been in that class in 2-3 years prior, and not everything sticks or gets built on).

What we tutors ended up doing most of the time was doing a lot of what you suggest -- winging it obviously, but in the process showing how to go about figuring out a tough math problem. So we'd ask them what they'd learned about in class that week (often a good clue as to what theorems might be being exercised) and basically had them get out their textbooks and we'd look through the section they were in to find something relevant to the problem. And yes, sometimes we'd get a student started on an approach that seemed like it would work but ended up not working out. And sometimes we had to call over another tutor to help us figure things out. At first I felt really bad about having to do that instead of just "knowing" what to tell them, but eventually I realized that we were modeling the type of thinking and problem solving they'd need to learn to do to be able to solve the problems on their own.

Back to your question:

In the interest of time, I wouldn't suggest lecturers come to class unprepared. However, I would suggest that maybe your preparation consist of "winging it" in advance, and keeping track of your approach, your wild goose chases, etc. and presenting a somewhat cleaned up summary of "how I solved this problem" in lecture. You can share some of your wrong turns, maybe in the context of, "at first I thought I could try X -- can anyone see why that wouldn't work?" (and if necessary give a quick summary of why that didn't work). That mitigates the issue of you wasting too much time on unproductive asides, while still helping them see what kinds of things you thought about when trying to solve the problem (or prove the theorem, or whatever you're demonstrating). And it saves the problem for students about notes being unintelligible because you can announce in advance that you're going to go off on an unproductive tangent, when you choose to do so for illustrative purposes.

It has happened to me too. I believe it leaves the impression that you didn't prepare properly (guilty as charged) and doesn't really help understanding. What does seems to work is to have the class make up a problem (or when they propose an interesting sidetrack) and solve that with the required backtracking, asking the class for input. Just making sure they know what is exploratory (I write "scratch" and use another color for this, perhaps on a separate area of the board) while/before writing up the definitive version. If it is interesting/important enough, or no satisfactory solution is reached, I'll write ìt up formally (scratch work included) to publish for them.

$\begingroup$An interesting idea. I occasionally get a chance to do something like this in homework problem sessions, when a student poses an extra question related to one of the HW problems.$\endgroup$
– Jyrki LahtonenApr 14 '14 at 13:25

With regard to the education literature (note that this is not specific to mathematics education) Keith Sawyer writes about the paradoxical balance of improvising in the classroom and having structured scripts in mind with regard to what is to be covered.

He also writes about teaching as performance, teacher expertise, and creativity in teaching, and certain paradoxes that emerge: the teacher paradox, the learning paradox, and the curriculum paradox.

Roughly speaking, the idea is that expert teachers are able to bring with them enough "scripts" so that even when there are mistakes or instances in which the topics covered in a class meeting veer off course, the teacher is sufficiently prepared to improvise in a way that leads to positive learning outcomes.

Sawyer writes a short introduction covering these topics for his book, with citation:

Sawyer, R. K. (2011). What makes good teachers great? The artful balance of structure and improvisation. Structure and improvisation in creative teaching, 1-24.

The intended take-away is that improvisation need not imply a lack of preparation; this is consistent with answers already provided in this thread, which I enjoyed and thought would be well-complemented by pointing interested readers to an additional reference.

(As for improvisation without preparation: This is lazy at best, and probably pedagogically damaging.)

When I am teaching a course for the second (or third or fourth or …) time, I pick problems that I know that students will get wrong, or wrong approaches that I know that they will take, and discuss them in class. After some student feedback indicated that this was confusing, I introduced the tactic of switching to red when I would write an intentionally wrong statement on the board. I also say that I will be doing this and that it will be signalled by the change to red, and I try to give a brief explanation of why I think that it is important.

I enjoy this approach, and wish in retrospect that my own teachers had implemented something like it—I believe that a trained mathematician's first two questions about any new definition are "What is an example?" and "What is a non-example?", and suspect that any scientist will ask similar questions. However, feedback about it, from students at all undergraduate levels, has ranged from neutral (no comment) to viciously negative. Perhaps the most common comment is "We don't want to learn the wrong way to do something." I do not remember that I have ever received a positive comment specifically mentioning this method.

I believe that this feedback is based on incorrect perceptions of what a mathematics class should be, and further that the long-term benefit will outweigh the negative student perceptions, so I have continued to implement the method. However, it is maybe worth knowing in advance the extent of the push-back that you might receive! (I should mention that I have not taught an undergraduate course targeted at math majors since I implemented this policy, so it may be that I am just seeing the perspective of non-majors on this approach.)

$\begingroup$Thank you for sharing. I have not tried using red for this purpose. Largely because until very recently I was restricted to blackboard and white chalk. But want to try this at some point.$\endgroup$
– Jyrki LahtonenAug 30 '14 at 5:45

1

$\begingroup$Late comment: You might try ju-jitsuing this into a series of "true or false?" exercises. My students recognize that as an expected technique.$\endgroup$
– Daniel R. CollinsFeb 11 '17 at 21:41

I love this approach, and I use it consistently. Particularly during problem solving.

Keep in mind, I teach using online videos. The feedback I get is 99.9% positive (the negative comments being largely weird, unserious comments).

Particularly, I get good feedback from students saying that they are motivated by realizing that teachers aren't, in fact, robots whose sole purpose is to repeat the text book. They are human beings, capable of making mistakes. And seeing your teacher make mistakes reminds you of this. It also shows how to bounce back from errors, and critiquing your own work when the answer seems blatantly wrong.

$\begingroup$I can see how this might indeed work better in an online format, if only because you can edit your videos afterwards if you find yourself getting too sidetracked.$\endgroup$
– Ilmari KaronenApr 14 '14 at 1:06

Everyone makes mistakes. I think the value in showing mistakes and talking through them is important not only to illustrate that you can be an intelligent, smart mathematician and still make mistakes... rather, the value in discussing mistakes is the learning opportunity involved related to developing an understanding of the content. Discussing why the mistake doesn't work and doesn't make sense is a useful counter example to compliment that examples that do work and do make sense.

I agree that the selected best answer for this question is excellent. I'd like to add that in Stephen Krantz's How to Teach Mathematics (AMS, 3rd Ed.), the primary thesis of the entire book is basically make sure not to do this. (In fact, conceivably the person who posed the question heard about it from this book.) Quotes from Sec. 1.2, "Prepare":

Some people rationalize not teaching well by saying (either to
themselves or to others), "My time is too valuable. I am not going to
spend it preparing my calculus lecture. I am so smart that I can just
walk into the classroom and wing it. And the students will benefit
from watching a mathematician think on his feet." (As a student, I
actually had professors who announced this nonsense to the class on a
regular basis. And, as you can imagine, these were professors who
royally botched up their lectures on a regular basis.)

It is true that most of us can walk into the room most of the time and
mostly wing it. But most of us will not be very successful if we do
so. Thirty minutes can be sufficient time for an experienced
instructor to prepare a calculus lecture... Make sure that you have
the definitions and theorems straight. It is a good idea to have a
single page of notes containing the key points and also briefly
listing your motivating ideas...

I cannot emphasize too strongly the fact that preparation is of the
utmost importance if you are going to deliver a stimulating class.
However it is also true that the more you prepare the more you lose
your spontaneity. You must strike a balance between (i) knowing the
material cold and (ii) being able to "talk things through" with your
audience.

I only teach smaller groups, but have found that structuring my exercises so that I deliberately put a trap or two in there for the students to discover and learn from is extremely beneficial.

The trick is to make sure the trap is small and (relatively) ineffective and provide the clues they need to find their own way out without being too obvious. One way of providing those clues is to solve a similar category of problem for them and emphasise certain key processes I follow to get back on track. It might be that I notice someone has a problem and I provide the answer they need while talking to another student nearby.

You need your students to discover the solutions themselves, but it took the maths community many years to make the discoveries that you want to teach them in a few weeks/months, so you also need them to remain resourceful and provide the clues close enough so they can join them together.

So making mistakes as a lecturer is useful where your students mentally copy your recovery strategies. (teaching them how you identify/recognise that you have gone off track, and what the heck you do to get back on a more productive path, how did you know it was just a messed up sign, or a totally different approach was required)

Bad idea. It's not such a radical idea that people can make mistakes that the students need to be exposed to it. Except for the next Fields medalists as a child, all of your students, even the strong ones, will have plenty of experience with making mistakes.

Now, giving some helpful tips on places where it is easy to go wrong is very helpful (like the Dummies book with the watch out symbols). But here, note that it is not just important to be strong at MATH, you have to be strong at KNOWING common student mistakes. And this is something that mostly comes from experience, from being open to experience and tabulating it, and perhaps discussion with other experienced teachers (who are real teachers, not just mathematicians). You can do cute things like coming up with nicknames for common errors (forgetting the constant, say). Or use red chalk for the watch out tips. But on purpose derailing the lecture (or even worse showing up unprepared and derailing by accident) is not a good move.

Understanding "how research mathematicians research" is crushingly irrelevant to most students through junior year of math classes (i.e. ALL service classes in uni and all HS math classes, and much of the ug math classes). Don't think this is some wonderful gift you are sharing. They will probably learn common issues of problem solving and research in the workplace more from their engineering co-op jobs. For the math undergrad, they can have a course on math research methods and do a senior project. Good enough.